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Problem solving strategies engel ; the role of critical thinking in problem solving; ystematic innovation an introduction to triz theory of inventive problem solvingystematic innovation an introduction to triz theory of inventive problem solving
Problem-Solving Strategies Arthur Engel Springer Arthur Engel Problem-Solving Strategies With 223 Figures 13 Angel Engel Institut făur Didaktik der Mathematik Johann Wolfgang GoetheUniversităat Frankfurt am Main Senckenberganlage 911 60054 Frankfurt am Main 11 Germany Series Editor: Paul R Halmos Department of Mathematics Santa Clara University Santa Clara, CA 95053 USA Mathematics Subject Classification (1991): 00A07 Library of Congress Cataloging-in-Publication Data Engel, Arthur Problem-solving strategies/Arthur Engel p cm — (Problem books in mathematics) Includes index ISBN 0-387-98219-1 (softcover: alk paper) Problem solving I Title II Series QA63.E54 1997 97-10090 510 76—dc21 © 1998 Springer-Verlag New York, Inc All rights reserved This work may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerpts in connection with reviews or scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed is forbidden The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, as understood by the Trade Marks and Merchandise Marks Act, may accordinly be used freely by anyone ISBN 0–387–98219–1 Springer-Verlag New York Berlin Heidelburg SPIN 10557554 Preface This book is an outgrowth of the training of the German IMO team from a time when we had only a short training time of 14 days, including half-day tests This has forced upon us a training of enormous compactness “Great Ideas” were the leading principles A huge number of problems were selected to illustrate these principles Not only topics but also ideas were efficient means of classification For whom is this book written? • For trainers and participants of contests of all kinds up to the highest level of international competitions, including the IMO and the Putnam Competition • For the regular high school teacher, who is conducting a mathematics club and is looking for ideas and problems for his/her club Here, he/she will find problems of any level from very simple ones to the most difficult problems ever proposed at any competition • For high school teachers who want to pose the problem of the week, problem of the month, and research problems of the year This is not so easy Many fail, but some persevere, and after a while they succeed and generate a creative atmosphere with continuous discussions of mathematical problems • For the regular high school teacher, who is just looking for ideas to enrich his/her teaching by some interesting nonroutine problems • For all those who are interested in solving tough and interesting problems The book is organized into chapters Each chapter starts with typical examples illustrating the main ideas followed by many problems and their solutions The vi Preface solutions are sometimes just hints, giving away the main idea leading to the solution In this way, it was possible to increase the number of examples and problems to over 1300 The reader can increase the effectiveness of the book even more by trying to solve the examples The problems are almost exclusively competition problems from all over the world Most of them are from the former USSR, some from Hungary, and some from Western countries, especially from the German National Competition The competition problems are usually variations of problems from journals with problem sections So it is not always easy to give credit to the originators of the problem If you see a beautiful problem, you first wonder at the creativity of the problem proposer Later you discover the result in an earlier source For this reason, the references to competitions are somewhat sporadic Usually no source is given if I have known the problem for more than 25 years Anyway, most of the problems are results that are known to experts in the respective fields There is a huge literature of mathematical problems But, as a trainer, I know that there can never be enough problems You are always in desperate need of new problems or old problems with new solutions Any new problem book has some new problems, and a big book, as this one, usually has quite a few problems that are new to the reader The problems are arranged in no particular order, and especially not in increasing order of difficulty We not know how to rate a problem’s difficulty Even the IMO jury, now consisting of 75 highly skilled problem solvers, commits grave errors in rating the difficulty of the problems it selects The over 400 IMO contestants are also an unreliable guide Too much depends on the previous training by an ever-changing set of hundreds of trainers A problem changes from impossible to trivial if a related problem was solved in training I would like to thank Dr Manfred Grathwohl for his help in implementing various LaTEX versions on the workstation at the institute and on my PC at home When difficulties arose, he was a competent and friendly advisor There will be some errors in the proofs, for which I take full responsibility, since none of my colleagues has read the manuscript before Readers will miss important strategies So I, but I have set myself a limit to the size of the book Especially, advanced methods are missing Still, it is probably the most complete training book on the market The gravest gap is the absence of new topics like probability and algorithmics to counter the conservative mood of the IMO jury One exception is Chapter 13 on games, a topic almost nonexistent in the IMO, but very popular in Russia Frankfurt am Main, Germany Arthur Engel Contents Preface v Abbreviations and Notations ix The Invariance Principle Coloring Proofs 25 The Extremal Principle 39 The Box Principle 59 Enumerative Combinatorics 85 Number Theory 117 Inequalities 161 The Induction Principle 205 Sequences 221 10 Polynomials 245 11 Functional Equations 271 viii Contents 12 Geometry 289 13 Games 361 14 Further Strategies 373 References 397 Index 401 Abbreviations and Notations Abbreviations ARO Allrussian Mathematical Olympiad ATMO Austrian Mathematical Olympiad AuMO Australian Mathematical Olympiad AUO Allunion Mathematical Olympiad BrMO British Mathematical Olympiad BWM German National Olympiad BMO Balkan Mathematical Olympiad ChNO Chinese National Olympiad HMO Hungarian Mathematical Olympiad (K˝urschak Competition) IIM International Intellectual Marathon (Mathematics/Physics Competition) IMO International Mathematical Olympiad LMO Leningrad Mathematical Olympiad MMO Moskov Mathematical Olympiad PAMO Polish-Austrian Mathematical Olympiad ... Cataloging-in-Publication Data Engel, Arthur Problem- solving strategies/ Arthur Engel p cm — (Problem books in mathematics) Includes index ISBN 0-387-98219-1 (softcover: alk paper) Problem solving I Title II... of new problems or old problems with new solutions Any new problem book has some new problems, and a big book, as this one, usually has quite a few problems that are new to the reader The problems... present our first Higher Problem- Solving Strategy It is extremely useful in solving certain types of difficult problems, which are easily recognizable We will teach it by solving problems which use